How do you call an object $X$ for which every monomorphism $i : X \hookrightarrow Y$ has a retract (i.e.\ a morphism $r : Y \rightarrow X$ such that $r \cdot i = 1_X$)?
I think of Y as an extension of X, and the retract as a way to show that X does not have extensions that provide new information, in that sense X is 'full' or 'maximal' or 'saturated', but all those terms are overloaded, and probably the notion has been defined somewhere in literature already. Any references? Thanks!
It is an Absolute Retract. (Copying a comment made by Qiaochu Yuan to enable registering it as an actual answer)